### Revision of the rest of the paper.

 ... ... @@ -79,7 +79,7 @@ A \emph{touching representation by comparable boxes} of a graph $G$ is a touchin such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. Let the \emph{comparable box dimension} $\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$. Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details. Then for a class $\GG$ of graphs, let $\cbdim(\GG):=\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the Then for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the comparable box dimension of graphs in $\GG$ is not bounded. Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular, ... ... @@ -126,21 +126,25 @@ Note that a clique with $2^d$ vertices has a touching representation by comparab where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$. Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$. In the following we consider the chromatic number $\chi(G)$, and one of its variants. A \emph{star coloring} of a graph $G$ is a proper coloring such that any two color classes induce a star forest (i.e., a graph not containing any 4-vertex path). The \emph{star chromatic number} $\chi_s(G)$ of $G$ is the minimum number of colors in a star coloring of $G$. We will need the fact that the star chromatic number is at most exponential in the comparable box dimension; this follows In the following we consider the chromatic number $\chi(G)$, and two of its variants. An \emph{acyclic coloring} (resp. \emph{star coloring}) of a graph $G$ is a proper coloring such that any two color classes induce a forest (resp. star forest, i.e., a graph not containing any 4-vertex path). The \emph{acyclic chromatic number} $\chi_a(G)$ (resp. \emph{star chromatic number} $\chi_s(G)$) of $G$ is the minimum number of colors in an acyclic (resp. star) coloring of $G$. We will need the fact that all the variants of the chromatic number are at most exponential in the comparable box dimension; this follows from~\cite{subconvex}, although we include an argument to make the dependence clear. \begin{lemma}\label{lemma-chrom} For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot 9^{\cbdim(G)}$. For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$, $\chi_a(G)\le 5^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot 9^{\cbdim(G)}$. \end{lemma} \begin{proof} We focus on the star chromatic number and note that the chromatic number may be bounded similarly. Suppose that $G$ has comparable box dimension $d$ witnessed by a representation $f$, and let $v_1, \ldots, v_n$ be the vertices of $G$ written so that $\vol(f(v_1)) \geq \ldots \geq \vol(f(v_n))$. Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy colouring $c$ so that $c(i)$ is the smallest color such that $c(i)\neq c(j)$ for any $jj$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1j$. Now define a greedy colouring $c$ so that $c(i)$ is the smallest color such that $c(i)\neq c(j)$ for any $jj$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$ ... ... @@ -516,14 +520,12 @@ clique-sums. \end{itemize} \end{proof} The following lemma shows that any graphs has a $C^\star$-clique-sum extendable representation in $\mathbb{R}^d$, for $d= \omega(G) + \ecbdim(G)$ and for any clique $C^\star$. The following lemma enables us to pick the root clique at the expense of increasing the dimension by $\omega(G)$. \begin{lemma}\label{lem-apex-cs} For any graph $G$ and any clique $C^\star$, we have that $G$ admits a $C^\star$-clique-sum extendable touching representation by comparabe boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus V(C^\star))$. For any graph $G$ and any clique $C^\star$, the graph $G$ admits a $C^\star$-clique-sum extendable touching representation by comparable boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus V(C^\star))$. \end{lemma} \begin{proof} The proof is essentially the same as the one of ... ... @@ -532,40 +534,43 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) + comparable boxes in$\mathbb{R}^{d'}$, with$d' = \cbdim(G\setminus V(C^\star))$, and let$V(C^\star) = \{v_1,\ldots,v_k\}$. We now construct the desired representation$h$of$G$as follows. For each vertex$v_i\in V(C^\star)$let$h(v_i)$be the box fulfilling (v1) with$d_{v_i} = i$. For each vertex$u\in V(G)\setminus V(C^\star)$, if$i\le k$then let$h(u)[i] = [0,1/2]$if$uv_i \in E(G)$, and$h(u)[i] = $v_i\in V(C^\star)$, let $h(v_i)$ be the box in $\mathbb{R}^d$ uniquely determined by the condition (v1) with $d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^\star)$, if $i\le k$ then let $h(u)[i] = [0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i] = [1/4,3/4]$ if $uv_i \notin E(G)$. For $i>k$ we have $h(u)[i] = \alpha_i h'(u)[i-k]$, for some $\alpha_i>0$. The values $\alpha_i>0$ are chosen suffciently small so that $h(u)[i] \subset [0,1)$, whenever $u\notin V(C^\star)$. \alpha h'(u)[i-k]$, for some$\alpha>0$. The value$\alpha>0$is chosen suffciently small so that$h(u)[i] \subset [0,1)$whenever$u\notin V(C^\star)$. We proceed similarly for the clique points. For any clique$C$of$G$, if$i\le k$then let$p(C)[i] = 0$if$v_i \in V(C)$, and$p(C)[i] = 1/4$if$v_i \notin V(C)$. For$i>k$we have to refer to the clique point$p'(C')$of$C'=C\setminus \{v_1,\ldots,v_k\}$, as we set$p(C)[i] = \alpha_i p'(C')[i-k]$. clique$C$of$G$, if$i\le k$then let$p(C)[i] = 0$if$v_i \in V(C)$, and$p(C)[i] = 1/4$if$v_i \notin V(C)$. For$i>k$we refer to the clique point$p'(C')$of$C'=C\setminus \{v_1,\ldots,v_k\}$, and we set$p(C)[i] = \alpha p'(C')[i-k]$. By the construction, it is clear that$h$is a touching representation of$G$. As$h'(u) \sqsubset h'(v)$implies that$h(u) \sqsubset h(v)$, and as$h(u) \sqsubset h(v_i)$, for every$u\in V(G)\setminus V(C^\star)$and every$v_i \in V(C^\star)$, we have that$h$is a touching representation by comparable boxes. By the construction, it is clear that$h$is a representation of$G$. For the$C^\star$-clique-sum extendability, it is clear that the (vertices) conditions hold. For the (cliques) condition (c1), let us first consider two distinct cliques$C_1$and$C_2h(u) \sqsubset h(v_i)$for every$u\in V(G)\setminus V(C^\star)$and every$v_i \in V(C^\star)$, we have that$h$is a representation by comparable boxes. For the$C^\star$-clique-sum extendability, the \textbf{(vertices)} conditions hold by the construction. For the \textbf{(cliques)} condition (c1), let us consider distinct cliques$C_1$and$C_2$of$G$such that$|V(C_1)| \ge |V(C_2)|$, and let$C'_i=C_i\setminus V(C^\star)$. If$C'_1 = C'_2$, there is a vertex$v_i \in V(C_1) \setminus V(C_2)$, and$p(C_1)[i] = 0 \neq 1/4 = p(C_2)[i]$. Otherwise, if$C'_1 \neq C'_2$, we have that$p'(C'_1) \neq p'(C'_2)$, which leads to Otherwise, if$C'_1 \neq C'_2$, then$p'(C'_1) \neq p'(C'_2)$, which implies$p(C_1) \neq p(C_2)$by construction. For the (cliques) condition (c2), let us first consider a vertex$v\in V(G)\setminus V(C^\star)$and a clique$C$of$G$containing$v$. In the first dimensions$i \le k$, we always have$h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if$v_i \in V(C)$we have$h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$(as in that case$v$and$v_i$are adjacent), and if$v_i \notin V(C)$we have$h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. Then for the last$d'$dimensions, by definition of$h'$, we have that$h^\varepsilon(C)[i] \subseteq h(v)[i]$for every$i>k$, except one, for which$h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. This completes the first case and we now consider a vertex$v\in V(G)\setminus V(C^\star)$and a clique$C$of$G$not containing$v$. As$v\notin V(C')$, there is an hyperplane${\mathcal H}' = \{ p\in \mathbb{R}^{d'}\ |\ p[i] = c\}$that separates$p'(C')$and$h'(v)$. This implies that the following hyperplane${\mathcal H} = \{ p\in \mathbb{R}^{d}\ |\ p[k+i] = \alpha_{k+i} c\}$separates$p(C)$and$h(v)$. Now we consider a vertex$v_i \in V(C^\star)$, and we note that for any clique$C$containing$v_i$For the \textbf{(cliques)} condition (c2), let us first consider a vertex$v\in V(G)\setminus V(C^\star)$and a clique$C$of$G$containing$v$. In the dimensions$i\in\{1,\ldots,k\}$, we always have$h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if$v_i \in V(C)$, then$h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$, as in this case$v$and$v_i$are adjacent; and if$v_i \notin V(C)$, then$h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. By the property (c2) of$h'$, we have$h^\varepsilon(C)[i] \subseteq h(v)[i]$for every$i>k$, except one, for which$h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. Next, let us consider a vertex$v\in V(G)\setminus V(C^\star)$and a clique$C$of$G$not containing$v$. As$v\notin V(C')$, the condition (c2) for$h'$implies that$p'(C')$is disjoint from$h'(v)$, and thus$p(C)$is disjoint from$h(v)$. Finally, we consider a vertex$v_i \in V(C^\star)$. Note that for any clique$C$containing$v_i$, we have that$h^\varepsilon(C)[i] \cap h(v_i)[i] = [0,\varepsilon]\cap [-1,0] = \{0\}$, and$h^\varepsilon(C)[j] \subseteq [0,1] = h(v_i)[j]$for any$j\neq i$. For a clique$C$that does not contain$v_i$we have that$h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$. ... ... @@ -578,184 +583,213 @@ of$\cbdim(G)$and$\chi(G)$. For any graph$G$,$\ecbdim(G) \le \cbdim(G) + \chi(G)$. \end{lemma} \begin{proof} Let$h$be a touching representation by comparable boxes of$G$in Let$h$be a touching representation of$G$by comparable boxes in$\mathbb{R}^d$, with$d=\cbdim(G)$, and let$c$be a$\chi(G)$-coloring of$G$. We start with a slightly modified version of$h$. We first scale$h$to fit in$(0,1)^d$, and for a sufficiently small real$\alpha>0$we increase each box in$h$, by sufficiently small real$\alpha>0$we increase each box in$h$by$2\alpha$in every dimension, that is we replace$h(v)[i] = [a,b]$by$[a-\varepsilon,b+\varepsilon]$for each vertex$v$and dimension$i$. Furthermore$\alpha$is chosen sufficiently small, so that no new intersection was created. The obtained representation$h_1$is thus an intersection representation of the same graph$G$such that, for every clique$C$of$G$, the intersection$I_C= \cap_{v\in V(C) h_1(v)}$is$d$-dimensional. For any maximal clique$C$of$G$, let$p_1(C)$be a point in the interior of$I_C$. by$[a-\alpha,b+\alpha]$for each vertex$v$and dimension$i$. We choose$\alpha$sufficiently small so that the boxes representing non-adjacent vertices remain disjoint, and thus the resulting representation$h_1$is an intersection representation of the same graph$G$. Moreover, observe that for every clique$C$of$G$, the intersection$I_C=\bigcap_{v\in V(C)} h_1(v)$is a box with non-zero edge lengths. For any clique$C$of$G$, let$p_1(C)$be a point in the interior of$I_C$different from the points chosen for all other cliques. Now we add$\chi(G)$dimensions to make the representation touching again, and to ensure some space for the clique boxes$h^\varepsilon(C)$. Formally we define$h_2$as follows. $$h_2(u)[i]=\begin{cases} h_1(u)[i]&\text{ if i\le d}\\ [1/5,3/5]&\text{ if c(u) < i-d}\\ [0,2/5]&\text{ if c(u) = i-d}\\ [1/5,3/5]&\text{ if i>d and c(u) < i-d}\\ [0,2/5]&\text{ if i>d and c(u) = i-d}\\ [2/5,4/5]&\text{ otherwise (if c(u) > i-d > 0)} \end{cases}$$ For any clique$C'$of$G$, let us denote$c(C')$, the color set$\{c(u)\ |\ u\in V(C')\}$, and let$C$be one of the maximal cliques containing$C'$. We now set $$p_2(C')[i]=\begin{cases} For any clique C of G, let c(C) denote the color set \{c(u)\ |\ u\in V(C)\}. We now set$$p_2(C)[i]=\begin{cases} p_1(C) &\text{ if$i\le d$}\\ 2/5 &\text{ if$i-d \in c(C')$}\\%\text{if$\exists u\in V(C)$with$c(u) = i-d$}\\ 2/5 &\text{ if$i>d$and$i-d \in c(C)}\\ 1/2 &\text{ otherwise} \end{cases} $$As h_2 is an extension of h_1, and as for all the extra dimensions j (j>d) we have that 2/5 \in h_2(v)[j] for every vertex v, we have that h_2 is an intersection representation of G. To prove that it is touching consider two adjacent As h_2 is an extension of h_1, and as in each dimension j>d, h_2(v)[j] is an interval of length 2/5 containing the point 2/5 for every vertex v, we have that h_2 is an intersection representation of G by comparable boxes. To prove that it is touching consider two adjacent vertices u and v such that c(u)d) the length of h_2(v)[j] is 2/5 for every vertex v, we have that the boxes in h_2 are comparables boxes. For the \emptyset-clique-sum extendability, the (vertices) conditions clearly hold. For the (cliques) conditions, let us first note that the points p_1(C), defined for the maximum cliques, are necessarily distinct. This impies that two cliques C_1 and C_2, which clique points p_2(C_1) and p_2(C_2) are based on distinct maximum cliques, necessarily lead to distinct points. In the case that C_1 and C_2 belong to some maximal clique C, we have that c(C_1) \neq c(C_2) and this implies by construction that p_2(C_1) and p_2(C_2) are distinct. Thus (c1) holds. By construction of h_1, we have that if h_2^{\varepsilon}(C')[i] \cap h_2(v)[i] is non-empty for every i\le d, then we have that h_2^{\varepsilon}(C')[i] \subset h_2(v)[i] for every i\le d, and we have that v belongs to some maximal clique C containing C'. If v\notin V(C') note that p_2(C')[d+c(v)] = 1/2 \notin[0,2/5]=h_2(v)[d+c(v)], while if v\in V(C') we have that h_2^{\varepsilon}(C')[i] \subset [2/5,1/2+\varepsilon] \subset h_2(v)[i] for every dimension i>d, except if c(v)=i-d, and in that case h_2(v)[i] \cap h_2^{\varepsilon}(C')[i] = [0,2/5]\cap[2/5,2/5+\varepsilon] = \{2/5\}. We thus have that (c2) holds, and this concludes the proof of the lemma. and h_2(v)[d+c(u)] = [2/5,4/5]. For the \emptyset-clique-sum extendability, the \textbf{(vertices)} conditions are void. For the \textbf{(cliques)} conditions, since p_1 is chosen to be injective, the mapping p_2 is injective as well, implying that (c1) holds. Consider now a clique C in G and a vertex v\in V(G). If c(v)\not\in c(C), then h_2(v)[c(v)+d]=[0,2/5] and p_2(C)[c(v)+d]=1/2, implying that h_2^{\varepsilon}(C)\cap h_2(v)=\emptyset. If c(v)\in c(C) but v\not\in V(C), then letting v'\in V(C) be the vertex of color c(v), we have vv'\not\in E(G), and thus h_1(v) is disjoint from h_1(v'). Since p_1(C) is contained in the interior of h_1(v'), it follows that h_2^{\varepsilon}(C)\cap h_2(v)=\emptyset. Finally, suppose that v\in C. Since p_1(C) is contained in the interior of h_1(v), we have h_2^{\varepsilon}(C)[i] \subset h_2(v)[i] for every i\le d. For i>d distinct from d+c(v), we have p_2^{\varepsilon}(C)[i]\in\{2/5,1/2\} and [2/5,3/5]\subseteq h_2(v)[i], and thus h_2^{\varepsilon}(C)[i] \subset h_2(v)[i]. For i=d+c(v), we have p_2^{\varepsilon}(C)[i]=2/5 and h_2(v)[i]=[0,2/5], and thus h_2^{\varepsilon}(C)[i] \cap h_2(v)[i]=\{p_2^{\varepsilon}(C)[i]\}. Therefore, (c2) holds. \end{proof} Together, the lemmas from this section show that comparable box dimension is almost preserved by full clique-sums. \begin{corollary}\label{cor-csum} Let \GG be a class of graphs of chromatic number at most k. If \GG' is the class of graphs obtained from \GG by repeatedly performing full clique-sums, then$$\cbdim(\GG')\le \cbdim(\GG) + 2k.$$\end{corollary} \begin{proof} Suppose a graph G is obtained from G_1, \ldots, G_m\in\GG by performing full clique-sums. Without loss of generality, the labelling of the graphs is chosen so that we first perform the full clique-sum on G_1 and G_2, then on the resulting graph and G_3, and so on. Let C^\star_1=\emptyset and for i=2,\ldots,m, let C^\star_i be the root clique of G_i on which it is glued in the full clique-sum operation. By Lemmas~\ref{lem-ecbdim-cbdim} and \ref{lem-apex-cs}, G_i has a C_i^\star-clique-sum extendable touching representation by comparable boxes in \mathbb{R}^d, where d=\cbdim(\GG) + 2k. Repeatedly applying Lemma~\ref{lem-cs}, we conclude that \cbdim(G)\le d. \end{proof} By Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}, this gives the following bounds. \begin{corollary}\label{cor-csump} Let \GG be a class of graphs of comparable box dimension at most d. \begin{itemize} \item The class \GG' of graphs obtained from \GG by repeatedly performing full clique-sums has comparable box dimension at most d + 2\cdot 3^d. \item The class of graphs obtained from \GG by repeatedly performing clique-sums has comparable box dimension at most 625^d. \end{itemize} \end{corollary} \begin{proof} The former bound directly follows from Corllary~\ref{cor-csum} and the bound on the chromatic number from Lemma~\ref{lemma-chrom}. For the latter one, we need to bound the star chromatic number of \GG'. Suppose a graph G is obtained from G_1, \ldots, G_m\in\GG by performing full clique-sums. For i=1,\ldots, m, suppose G_i has an acyclic coloring \varphi_i by at most k colors. Note that the vertices of any clique get pairwise different colors, and thus by permuting the colors, we can ensure that when we perform the full clique-sum, the vertices that are identified have the same color. Hence, we can define a coloring \varphi of G such that for each i, the restriction of \varphi to V(G_i) is equal to \varphi_i. Let C be the union of any two color classes of \varphi. Then for each i, G_i[C\cap V(G_i)] is a forest, and since G[C] is obtained from these graphs by full clique-sums, G[C] is also a forest. Hence, \varphi is an acyclic coloring of G by at most k colors. By~\cite{albertson2004coloring}, G has a star coloring by at most 2k^2-k colors. Hence, Lemma~\ref{lemma-chrom} implies that \GG' has star chromatic number at most 2\cdot 25^d - 5^d. The bound on the comparable box dimension of subgraphs of graphs from \GG' then follows from Lemma~\ref{lemma-subg}. \end{proof} \section{The strong product structure and minor-closed classes} A \emph{k-tree} is any graph obtained by repeated full clique-sums on cliques of size k from cliques of size at most k+1. A \emph{k-tree-grid} is a strong product of a k-tree and a path. An \emph{extended k-tree-grid} is a graph obtained from a k-tree-grid by adding at most k apex vertices. Dujmovi{\'c} et al.~\cite{DJM+} proved the following result. \begin{theorem}\label{thm-prod} Any graph G is a subgraph of the strong product of a k-tree, a path, and K_m, where Any graph G is a subgraph of the strong product of a k-tree-grid and K_m, where \begin{itemize} \item k=3 and m=3 if G is planar, and \item k=4 and m=\max(2g,3) if G has Euler genus at most g. \end{itemize} Moreover, for every t, there exists a k such that any K_t-minor-free graph G is a subgraph of a graph obtained from successive clique-sums of graphs, that are obtained from the strong product of a path and a k-tree, by adding at most k apex vertices. Moreover, for every t, there exists an integer k such that any K_t-minor-free graph G is a subgraph of a graph obtained by repeated clique-sums from extended k-tree-grids. \end{theorem} Let us first bound the comparable box dimension of a graph in terms of its Euler genus. As paths and m-cliques admit touching representations with hypercubes of unit size in \mathbb{R}^{1} and in \mathbb{R}^{\lceil \log_2 m \rceil} respectively, by Lemma~\ref{lemma-sp} it suffice to bound the comparable box dimension of k-trees. Lemma~\ref{lemma-sp} it suffices to bound the comparable box dimension of k-trees. \begin{theorem}\label{thm-ktree} For any k-tree G, \cbdim(G) \le \ecbdim(G) \le k+1. \end{theorem} \begin{proof} Note that there exists a k-tree G' having a k-clique C^\star such that G'\setminus V(C^\star) corresponds to G. Let us construct a C^\star-clique-sum extendable representation of G' and note that it induces a \emptyset-clique-sum extendable representation of G. Note that G' can be obtained by starting with a (k+1)-clique containing C^\star, and by performing successive full clique-sums of K_{k+1} on a K_k subclique. By Lemma~\ref{lem-cs}, it suffice to show that K_{k+1}, the (k+1)-clique with vertex set \{v_1, \ldots, v_{k+1}\}, has a (K_{k+1} -\{v_{k+1}\})-clique-sum extendable touching representation by hypercubes. Let us define such touching representation h as follows: \begin{itemize} \item h(v_i)[i] = [-1,0] if i\le k \item h(v_i)[j] = [0,1] if i\le k and i\neq j \item h(v_{k+1})[j] = [0,\frac12] for any j \end{itemize} One can easily check that the (vertices) conditions are fulfilled. For the (cliques) conditions let us set Let H be a complete graph with k+1 vertices and let C^\star be a clique of size k in H. By Lemma~\ref{lem-cs}, it suffices to show that H has a C^\star-clique-sum extendable touching representation by hypercubes in \mathbb{R}^{k+1}. Let V(C^\star)=\{v_1,\ldots,v_k\}. We construct the representation h so that (v1) holds with d_{v_i}=i for each i; this uniquely determines the hypercubes h(v_1), \ldots, h(v_k). For the vertex v_{k+1} \in V(H)\setminus V(C^\star), we set h(v_{k+1})=[0,1/2]^{k+1}. This ensures that the \textbf{(vertices)} conditions holds. For the \textbf{(cliques)} conditions, let us set the point p(C) for every clique C as follows: \begin{itemize} \item p(C)[i] = 0 for every i\le k and if v_i\in C \item p(C)[i] = \frac14 for every i\le k and if v_i\notin C \item p(C)[i] = 0 for every i\le k such that v_i\in C \item p(C)[i] = \frac14 for every i\le k such that v_i\notin C \item p(C)[k+1] = \frac12 if v_{k+1}\in C \item p(C)[k+1] = \frac34 if v_{k+1}\notin C \end{itemize} By construction, it is clear that p(C) \in h(v_i) if and only if v_i\in V(C). Let us check the other (cliques) conditions. By construction, it is clear that for each vertex v\in V(H), p(C) \in h(v) if and only if v\in V(C). For any two distinct cliques C_1 and C_2 the points p(C_1) and p(C_2) are distinct. Indeed, if |V(C_1)|\ge |V(C_2)| there is a vertex v_i\in V(C_1)\setminus V(C_2), and this implies that p(C_1)[i] < p(C_2)[i]. For a vertex v_i and a clique C, the boxes h(v_i) and h^\varepsilon(C) intersect if and only if v_i\in V(C). Indeed, if v_i\in V(C) then p(C)\in h(v_i) and p(C)\in h^\varepsilon(C), and if v_i\notin V(C) then h(v_i)[i] = [-1,0] if i\le k (resp. h(v_i)[i] = [0, \frac12] if i= k+1) and h^\varepsilon(C)[i] = [\frac14,\frac14+\varepsilon] (resp. h^\varepsilon(C)[i] = [\frac34,\frac34+\varepsilon]). Finally, if v_i\in V(C) we have that h(v_i)[i] \cap h^\varepsilon(C)[i] = \{p(C)[i]\} and that h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon] for any j\neq i and any \varepsilon <\frac14. This concludes the proof of the theorem. p(C_2) are distinct. Indeed, by symmetry we can assume that for some i, we have v_i\in V(C_1)\setminus V(C_2), and this implies that p(C_1)[i] < p(C_2)[i]. Hence, the condition (v1) holds. Consider now a vertex v_i and a clique C. As we observed before, if v_i\not\in V(C), then p(C) \not\in h(v_i), and thus h^\varepsilon(C) and h(v_i) are disjoint (for sufficiently small \varepsilon>0). If v_i\in C, then the definitions ensure that p(C)[i] is equal to the maximum of h(v_i)[i], and that for j\neq i, p(C)[j] is in the interior of h(v_i)[j], implying h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon] for sufficiently small \varepsilon>0. \end{proof} The \emph{treewidth} \tw(G) of a graph G is the minimum k such that G is a subgraph of a k-tree. Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree} extends to graphs of treewidth at most k. \begin{corollary}\label{cor-tw} Every graph G satisfies \cbdim(G)\le\tw(G)+1. \end{corollary} \begin{proof} Let k=\tw(G). Observe that there exists a k-tree T with the root clique C^\star such that G\subseteq T-V(C^\star). Inspection of the proof of Theorem~\ref{thm-ktree} (and Lemma~\ref{lem-cs}) shows that we obtain a representation h of T-V(C^\star) in \mathbb{R}^{k+1} such that \begin{itemize} \item the vertices are represented by hypercubes of pairwise different sizes, \item if uv\in E(T-V(C^\star)) and h(u)\sqsubseteq h(v), then h(u)\cap h(v) is a facet of h(u) incident with its point with minimum coordinates, and \item for each vertex u and each facet of h(u) incident with its point with minimum coordinates, there exists at most one vertex v such that uv\in E(T-V(C^\star)) and h(u)\sqsubseteq h(v). \end{itemize} If for some u,v\in V(G), we have uv\in E(T)\setminus E(G), where without loss of generality h(u)\sqsubseteq h(v), we now alter the representation by shrinking h(u) slighly away from h(v) (so that all other touchings are preserved). Since the hypercubes of h have pairwise different sizes, the resulting touching representation of G is by comparable boxes. \end{proof} Note that actually the bound on the comparable boxes dimension of Theorem~\ref{thm-ktree} extends to graphs of treewidth k. For this, note that the construction in this proof can provide us with a representation h of any k-tree G with hypercubes of distinct sizes. Note also that this representation is such that for any two adjacent vertices u and v, with h(u) \sqsubset h(v) say, the intersection I = h(u) \cap h(v) is a facet of h(u). Actually I[i] = h(u)[i] for every dimension, except one that we denote j. For this dimension we have that I[j]=\{c\} for some c, and that h(u)[j]=[c,c+s], where s is the length of the sides of h(u). In that context to delete an edge uv one can simply replace h(u)[j]=[c,c+s] with [c+\varepsilon,c+s], for a sufficiently small \varepsilon. One can proceed similarly for any subset of edges, and note that as the hypercubes in h have distinct sizes these small perturbations give rise to boxes that are still comparable. Thus for any treewidth k graph H (that is a subgraph of a k-tree G) we have \cbdim(H)\le k+1. As every planar graph G has a touching representation by cubes in \mathbb{R}^3~\cite{felsner2011contact}, we have that \cbdim(G)\le 3. For the graphs with higher Euler genus we can also derive upper \mathbb{R}^3~\cite{felsner2011contact}, we have that \cbdim(G)\le 3. For the graphs with higher Euler genus we can also derive upper bounds. Indeed, combining the previous observation on the representations of paths and K_m, with Theorem~\ref{thm-ktree}, Lemma~\ref{lemma-sp}, and Corollary~\ref{cor-subg} we obtain: \begin{corollary}\label{cor-genus} For every graph G of Euler genus g, there exists a supergraph G' of G such that \cbdim(G')\le 5+1+\lceil \log_2 \max(2g,3)\rceil. Consequently,$$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.$$of G such that \cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil. Consequently,$$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.\end{corollary} Let us now finally prove Theorem~\ref{thm-minor}, using the structure provided by Theorem~\ref{thm-prod}. We have seen that the strong product of a path and a k-tree has bounded comparable boxes dimension, and by Lemma~\ref{lemma-apex} adding at most k apex vertices keeps the dimension bounded. Then by Lemma~\ref{lemma-chrom} and Lemma~\ref{lem-ecbdim-cbdim}, these graphs admit a \emptyset-clique-sum extendable representations in bounded dimensions. As the obtained graphs have bounded dimension, by Lemma~\ref{lemma-cliq} and Lemma~\ref{lem-apex-cs}, for any choice of a root clique C^\star, they have a C^\star-clique-sum extendable representation in bounded dimension. Thus by Lemma~\ref{lem-cs} any sequence of clique sum from these graphs leads to a graph with bounded dimension. Finally, we have seen that taking a subgraph does not lead to undounded dimension, and we obtain Theorem~\ref{thm-minor}. Similarly, we can deal with proper minor-closed classes. \begin{proof}[Proof of Theorem~\ref{thm-minor}] Let \GG be a proper minor-closed class. Since \GG is proper, there exists t such that K_t\not\in \GG. By Theorem~\ref{thm-prod}, there exists k such that every graph in \GG is a subgraph of a graph obtained by repeated clique-sums from extended k-tree-grids. As we have seen, k-tree-grids have comparable box dimension at most k+2, and by Lemma~\ref{lemma-apex}, extended k-tree-grids have comparable box dimension at most 2k+2. By Corollary~\ref{cor-csump}, it follows that \cbdim(\GG)\le 625^{2k+2}. \end{proof} Note that the graph obtained from K_{2n} by deleting a perfect matching has Euler genus \Theta(n^2) and comparable box dimension n. It follows that the dependence of the comparable box dimension cannot be and comparable box dimension n. It follows that the dependence of the comparable box dimension on the Euler genus cannot be subpolynomial (though the degree \log_2 81 of the polynomial established in Corollary~\ref{cor-genus} certainly can be improved). The dependence of the comparable box dimension on the size of the forbidden minor that we established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}. ... ... @@ -808,9 +842,7 @@ the smallest axis-aligned hypercube containing f(v), then there exists a posit (f,\omega) is an s_d-comparable envelope representation of G in \mathbb{R}^d of thickness at most 2. \end{itemize} \note{TO REMOVE if we reintroduce tree decomposition earlier !} \note{Let us recall some notions about treewidth. Let us recall some notions about treewidth. A \emph{tree decomposition} of a graph G is a pair (T,\beta), where T is a rooted tree and \beta:V(T)\to 2^{V(G)} assigns a \emph{bag} to each of its nodes, such that ... ... @@ -821,11 +853,9 @@ (f,\omega) is an s_d-comparable envelope representation of G in \mathbb{R non-empty and induces a connected subtree of T. \end{itemize} For nodes x,y\in V(T), we write x\preceq y if x=y or x is a descendant of y in T. %For each vertex v\in V(G), let p(v) be the node x\in V(T) such that v\in \beta(x) and x is nearest to the root of T. %The \emph{adhesion} of the tree decomposition is the maximum of |\beta(x)\cap\beta(y)| over distinct x,y\in V(T), %and its The \emph{width} of the tree decomposition is the maximum of the sizes of the bags minus 1. The \emph{treewidth} of a graph is the minimum of the widths of its tree decompositions.} of the widths of its tree decompositions. Let us remark that the value of treewidth obtained via this definition coincides with the one via k-trees which we used in the previous section. \begin{theorem}\label{thm-twfrag} For positive integers t, s, and d, the class of graphs ... ... @@ -872,7 +902,7 @@\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_ By the union bound, we conclude that\text{Pr}[v_a\in X]\le 1/k$. Let us now bound the treewidth of$G-X$. For$a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of$\mathbb{R}^d\setminus (\cup_{H\in \HH^a} H)$. For$a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of$\mathbb{R}^d\setminus \bigl(\bigcup_{H\in \HH^a} H\bigr)$. A set$C\subseteq\mathbb{R}^d$is a \emph{cell} if it is an$a$-cell for some$a\ge 0$. A cell$C$is \emph{non-empty} if there exists$v\in V(G-X)$such that$\iota(v)\subseteq C$. Note that there exists a rooted tree$T$whose vertices are ... ...  ... ... @@ -5350,3 +5350,11 @@ note = {In Press} year=1994, pages={133--138}} } @article{albertson2004coloring, title={Coloring with no$2 $-Colored$ P\_4 \$'s}, author={Albertson, Michael O and Chappell, Glenn G and Kierstead, Hal A and K{\"u}ndgen, Andr{\'e} and Ramamurthi, Radhika}, journal={the electronic journal of combinatorics}, pages={R26--R26}, year={2004} }